no
If he wants to.
Many times (In the movies and TV shows) Batman has threatened to kill The Joker. He has never carried out his threats because it would violate his conscience. There are many comics where Joker does die and I believe there is an alternative comic called 'The nail' where Batman kills The Joker.
Yes because joker killed Jason Todd the second robin
Yes, the joker did kill people in the movie Dark Knight.
The Joker.
The joker
Joker
Not in the comics universe.
No, the Joker from Batman is a fictional (non real) character.
When the Joker meets Batman the Joker calls him my dear Darknight.
No, but in the comics he sometimes loses it and tries to kill The Joker.